Integrand size = 15, antiderivative size = 126 \[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=-\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}+\frac {1}{6} \left (a+b \tan ^4(x)\right )^{3/2} \]
-1/2*(a+b)^(3/2)*arctanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))- 1/4*(3*a+2*b)*arctanh(b^(1/2)*tan(x)^2/(a+b*tan(x)^4)^(1/2))*b^(1/2)+1/4*( a+b*tan(x)^4)^(1/2)*(2*a+2*b-b*tan(x)^2)+1/6*(a+b*tan(x)^4)^(3/2)
Time = 4.82 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.32 \[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\frac {1}{12} \left (-6 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-6 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\sqrt {a+b \tan ^4(x)} \left (8 a+6 b-3 b \tan ^2(x)+2 b \tan ^4(x)\right )-\frac {3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \tan ^4(x)}}{\sqrt {1+\frac {b \tan ^4(x)}{a}}}\right ) \]
(-6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] - 6*( a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + Sqrt[a + b*Tan[x]^4]*(8*a + 6*b - 3*b*Tan[x]^2 + 2*b*Tan[x]^4) - (3*Sqrt [a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]^4])/Sqrt [1 + (b*Tan[x]^4)/a])/12
Time = 0.37 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 4153, 1577, 493, 682, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (x) \left (a+b \tan (x)^4\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int \frac {\tan (x) \left (a+b \tan ^4(x)\right )^{3/2}}{\tan ^2(x)+1}d\tan (x)\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (b \tan ^4(x)+a\right )^{3/2}}{\tan ^2(x)+1}d\tan ^2(x)\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {b \left (a (2 a+b)-b (3 a+2 b) \tan ^2(x)\right )}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{2 b}+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {a (2 a+b)-b (3 a+2 b) \tan ^2(x)}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-b (3 a+2 b) \int \frac {1}{\sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-b (3 a+2 b) \int \frac {1}{1-b \tan ^4(x)}d\frac {\tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}\right )+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\right )+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-2 (a+b)^2 \int \frac {1}{-\tan ^4(x)+a+b}d\frac {a-b \tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\right )+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-2 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\right )+\frac {1}{3} \left (a+b \tan ^4(x)\right )^{3/2}+\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
((-(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]) - 2*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^ 4])])/2 + ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/2 + (a + b*Tan[x ]^4)^(3/2)/3)/2
3.4.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(103)=206\).
Time = 0.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.94
| method | result | size |
| derivativedivides | \(-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2}+\frac {b^{2} \left (\frac {\tan \left (x \right )^{4} \sqrt {a +b \tan \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \tan \left (x \right )^{4}}}{3 b^{2}}\right )}{2}+\frac {b \sqrt {a +b \tan \left (x \right )^{4}}}{2}-a \sqrt {b}\, \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )-\frac {b^{2} \left (\frac {\tan \left (x \right )^{2} \sqrt {a +b \tan \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}+a \sqrt {a +b \tan \left (x \right )^{4}}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \sqrt {a +b}}\) | \(245\) |
| default | \(-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2}+\frac {b^{2} \left (\frac {\tan \left (x \right )^{4} \sqrt {a +b \tan \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \tan \left (x \right )^{4}}}{3 b^{2}}\right )}{2}+\frac {b \sqrt {a +b \tan \left (x \right )^{4}}}{2}-a \sqrt {b}\, \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )-\frac {b^{2} \left (\frac {\tan \left (x \right )^{2} \sqrt {a +b \tan \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}+a \sqrt {a +b \tan \left (x \right )^{4}}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \sqrt {a +b}}\) | \(245\) |
-1/2*b^(3/2)*ln(b^(1/2)*tan(x)^2+(a+b*tan(x)^4)^(1/2))+1/2*b^2*(1/3*tan(x) ^4/b*(a+b*tan(x)^4)^(1/2)-2/3*a/b^2*(a+b*tan(x)^4)^(1/2))+1/2*b*(a+b*tan(x )^4)^(1/2)-a*b^(1/2)*ln(b^(1/2)*tan(x)^2+(a+b*tan(x)^4)^(1/2))-1/2*b^2*(1/ 2*tan(x)^2/b*(a+b*tan(x)^4)^(1/2)-1/2*a/b^(3/2)*ln(b^(1/2)*tan(x)^2+(a+b*t an(x)^4)^(1/2)))+a*(a+b*tan(x)^4)^(1/2)-1/2*(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln ((2*a+2*b-2*b*(1+tan(x)^2)+2*(a+b)^(1/2)*(b*(1+tan(x)^2)^2-2*b*(1+tan(x)^2 )+a+b)^(1/2))/(1+tan(x)^2))
Time = 0.45 (sec) , antiderivative size = 593, normalized size of antiderivative = 4.71 \[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\left [\frac {1}{8} \, {\left (3 \, a + 2 \, b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + \frac {1}{4} \, {\left (a + b\right )}^{\frac {3}{2}} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{12} \, {\left (2 \, b \tan \left (x\right )^{4} - 3 \, b \tan \left (x\right )^{2} + 8 \, a + 6 \, b\right )} \sqrt {b \tan \left (x\right )^{4} + a}, \frac {1}{4} \, {\left (3 \, a + 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{4} \, {\left (a + b\right )}^{\frac {3}{2}} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{12} \, {\left (2 \, b \tan \left (x\right )^{4} - 3 \, b \tan \left (x\right )^{2} + 8 \, a + 6 \, b\right )} \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, {\left (a + b\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + \frac {1}{8} \, {\left (3 \, a + 2 \, b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + \frac {1}{12} \, {\left (2 \, b \tan \left (x\right )^{4} - 3 \, b \tan \left (x\right )^{2} + 8 \, a + 6 \, b\right )} \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, {\left (a + b\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + \frac {1}{4} \, {\left (3 \, a + 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{12} \, {\left (2 \, b \tan \left (x\right )^{4} - 3 \, b \tan \left (x\right )^{2} + 8 \, a + 6 \, b\right )} \sqrt {b \tan \left (x\right )^{4} + a}\right ] \]
[1/8*(3*a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b )*tan(x)^2 - a) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*ta n(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b )/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^ 4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x )^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/12*(2*b*tan(x)^4 - 3*b*ta n(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), -1/2*(a + b)*sqrt(-a - b)*arctan (sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/8*(3*a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x )^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), -1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x) ^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^ 2)) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a)]
\[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \tan {\left (x \right )}\, dx \]
\[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \tan \left (x\right ) \,d x } \]
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10 \[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (3 \, a \sqrt {b} + 2 \, b^{\frac {3}{2}}\right )} \log \left ({\left | -\sqrt {b} \tan \left (x\right )^{2} + \sqrt {b \tan \left (x\right )^{4} + a} \right |}\right ) + \frac {1}{12} \, \sqrt {b \tan \left (x\right )^{4} + a} {\left ({\left (2 \, b \tan \left (x\right )^{2} - 3 \, b\right )} \tan \left (x\right )^{2} + \frac {2 \, {\left (4 \, a b + 3 \, b^{2}\right )}}{b}\right )} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (-\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b}} \]
1/4*(3*a*sqrt(b) + 2*b^(3/2))*log(abs(-sqrt(b)*tan(x)^2 + sqrt(b*tan(x)^4 + a))) + 1/12*sqrt(b*tan(x)^4 + a)*((2*b*tan(x)^2 - 3*b)*tan(x)^2 + 2*(4*a *b + 3*b^2)/b) + (a^2 + 2*a*b + b^2)*arctan(-(sqrt(b)*tan(x)^2 - sqrt(b*ta n(x)^4 + a) + sqrt(b))/sqrt(-a - b))/sqrt(-a - b)
Timed out. \[ \int \tan (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int \mathrm {tan}\left (x\right )\,{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{3/2} \,d x \]